The smallest square number divisible by 6, 9, and 15 is 900.
To find this, we first need to determine the least common multiple (LCM) of 6, 9, and 15.
- Prime factorization of 6: 2 x 3
- Prime factorization of 9: 3 x 3 (or 32)
- Prime factorization of 15: 3 x 5
The LCM is found by taking the highest power of each prime factor present in the factorizations: 21 x 32 x 51 = 2 x 9 x 5 = 90.
However, 90 is not a perfect square. To make it a perfect square, each prime factor must have an even exponent. Currently, the prime factorization of 90 (2 x 32 x 5) has prime factors 2 and 5 raised to the power of 1. We need to multiply 90 by 2 and 5 to give them an exponent of 2.
So, we multiply 90 by (2 x 5) = 10.
Therefore, 90 x 10 = 900.
The prime factorization of 900 is 22 x 32 x 52, which is (2 x 3 x 5)2 = 302. Since 900 = 302, it is a square number. It is also divisible by 6, 9, and 15.
